HMMT 二月 2012 · 代数 · 第 3 题

HMMT February 2012 — Algebra — Problem 3

Given points a and b in the plane, let a ⊕ b be the unique point c such that abc is an equilateral triangle with a, b, c in the clockwise orientation. Solve ( x ⊕ (0 , 0)) ⊕ (1 , 1) = (1 , − 1) for x .

解析

Given points a and b in the plane, let a ⊕ b be the unique point c such that abc is an equilateral triangle with a, b, c in the clockwise orientation. Solve ( x ⊕ (0 , 0)) ⊕ (1 , 1) = (1 , − 1) for x . √ √ 1 − 3 3 − 3 Answer: ( , ) It is clear from the definition of ⊕ that b ⊕ ( a ⊕ b ) = a and if a ⊕ b = c 2 2 √ then b ⊕ c = a and c ⊕ a = b . Therefore x ⊕ (0 , 0) = (1 , 1) ⊕ (1 , − 1) = (1 − 3 , 0). Now this means √ √ √ 1 − 3 3 − 3 x = (0 , 0) ⊕ (1 − 3 , 0) = ( , ). 2 2